Israeli attacks killed at least 31 people and wounded 40 others in Lebanon on Tuesday, according to the Lebanese Ministry of Public Health, as Israel said it was "deepening" its military operation there.

Four children were among the dead and 10 other children were among the injured, the health ministry said.

Since March 2, Israeli forces have killed at least 3,213 people and wounded another 9,737 across Lebanon, according to the latest tally from the health ministry. The continued killings come despite a U.S.-brokered ceasefire agreed between Lebanese and Israeli leaders in April.

Israel claims its strikes are targeting Hezbollah fighters and infrastructure, but Lebanese officials have reported many civilians among the casualties.

On Wednesday, Israel Defense Forces spokesperson Avichay Adraee issued a new evacuation order for the southern Lebanese city of Nabatieh, warning that the IDF intended to act "forcefully" against alleged Hezbollah targets in the area.

Hezbollah, meanwhile, continues its attacks on Israeli forces prosecuting Israel's ground campaign inside southern Lebanon. The Iran-backed group is also regularly launching longer-range attacks into northern Israel.

-ABC News' Morgan Winsor

Vice President JD Vance told NBC News in an interview published on Tuesday that he feels "extremely hopeful" the country will agree to a deal that includes a commitment not to develop nuclear weapons.

Vance's comments follow Iran's accusation that the U.S. breached the ceasefire between the two countries after U.S. Central Command said it carried out "self-defense" strikes on Monday.

"I think the more difficult question," Vance said, "is whether they agree to the kind of enforcement mechanism, the kind of monitoring mechanism, that gives us confidence that they won't violate the deal in the future."

Vance also said that his faith has informed his thinking throughout the conflict.

Vance said that "just war" theory forces leaders "to ask very difficult questions about whether a war is justified."

"But at its best," Vance said, "it's forcing you to ask the right questions. So, I find myself constantly asking myself, 'Is this justified? Is this moral? Is this the right thing to do?' And that does provide a limitation on political leaders, as it should."

-ABC News' Hannah Demissie

Monday's strikes in Iran were prompted by the U.S. detecting a pattern of increasing and potentially threatening Iranian activity over a 24-hour period, according to a U.S. official.

U.S. Central Command had said Monday that U.S. forces "conducted self-defense strikes in southern Iran to protect our troops from threats posed by Iranian forces."

"Targets included missile launch sites and Iranian boats attempting to emplace mines," it said.

-ABC News' Luis Martinez

Despite the latest U.S. strikes in Iran, talks toward an initial agreement to end the war continued in Doha on Tuesday, with a high-level Iranian delegation and Qatari mediators meeting in the city and the Trump administration’s negotiators participating from afar, according to two U.S. officials.

Officials and mediators say that Iran and the U.S. are working to reach an agreement on an initial memorandum of understanding that would end the conflict and gradually reopen the Strait of Hormuz while nuclear issues were negotiated.

Although the memo would allow for nuclear negotiations to take place over at least 60 days, the Trump administration still wants to see Iran agree to some big-picture commitments before signing onto the framework, U.S. officials say.

Iran, meanwhile, continues to try to maximize economic incentives to be included in the initial phase of the agreement -- including pushing for the release of frozen Iranian assets, the U.S. officials added.

Another factor is the situation in Lebanon. Iran -- which has insisted Israel's fight against Hezbollah was part of the initial ceasefire deal -- is adamant that the memo should end the conflict there. However, the Trump administration has so far indicated it will continue to back Israel's right to defend itself from Hezbollah.

-ABC News' Shannon K. Kingston